\[ \int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {55 a^{\frac {5}{2}} \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}}{\sqrt {a +a \sin \left (f x +e \right )}}\right )}{8 f}-\frac {2 a \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{5 f}-\frac {5 a \cot \left (f x +e \right ) \csc \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{12 f}-\frac {\cot \left (f x +e \right ) \left (\csc ^{2}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{3 f}-\frac {9 a^{3} \cos \left (f x +e \right )}{40 f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {16 a^{2} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{15 f}+\frac {17 a^{2} \cot \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{24 f} \]
command
integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (768 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 2560 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 825 \, \sqrt {2} a^{2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 1920 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \frac {20 \, {\left (108 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 176 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 69 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3}}\right )} \sqrt {a}}{480 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________